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Let  f(x) = (2x^4 - 8x^3 + x^2 + 7x - 3)/(x^2 + x - 2).  Give a polynomial g(x) so that f(x) + g(x) has a horizontal asymptote of 0 as x approaches positive infinity.

 Nov 15, 2020
 #1
avatar+114221 
+2

This seems trickier than it  really is.......

 

A rational function  will  have a horizontal asymptote of 0 whenever  the degree of the polynomial in the  numerator is less than the degree of the ploynomial in the denominator

 

So....all we need to have g(x)  be is something where we change the signs on the 2x^4, -8x^3  and x^2 terms in the original polynomial  (keeping the same  denominator)

 

So....let  g(x)    be   

 

(-2x^4  + 8x^3  - x^2 )

__________________

(x^2 + x - 2)

 

 

When we add this to f(x)  we get

 

    (7x - 3)

__________     

(x^2 + x - 2)

 

And this function will have a horizontal asymptote of 0  as x approaches infinity

 

 

cool cool cool

 Nov 15, 2020
 #2
avatar+112016 
+1

Nice work Chris.

 

You just need the degree of the numerator to be less than the degree of the denominator.

Which is exactly what CPhill did!    cool

 Nov 15, 2020
edited by Melody  Nov 15, 2020
 #3
avatar+2197 
+2

:0 Chris exists! Welcome back laugh

 Nov 15, 2020
 #4
avatar+114221 
+1

Thanks, Melody....and yeah, Cal....I DO exist   (but just barely....LOL!!!!  )

 

 

cool cool cool

 Nov 15, 2020
 #5
avatar+2197 
+1

"Just barely"? What do you mean angry Of course you exist!!

 Nov 16, 2020

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